Q) Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y) such that AB is a diameter of the circle. Find the value of y. Also, find the radius of the circle.

Ans:

(i) value of y:

We know that the center is the midpoint of the diameter,

Therefore if AB is the diameter and O is the centre of the circle,

then O will be midpoint of BD

and OA and OB will be equal and radii of the circle

Next, we know that the coordinates of midpoint are given by:

(X,Y) =

By substituting the given values, we get:

(2 , – 3y) =

By equating y coordinates, we get: – 3 y =

∴ – 3y x 2 = y + 7

∴ – 6 y = y + 7

∴ – 7 y = 7

∴ y = – 1

Therefore, the value of y = – 1

(ii) Radius of circle:

Radius of the circle is the length of line segment OA or OB

We have coordinates of Centre O (2, – 3 x -1) or (2, 3) and A (-1, -1)

We know that the distance between two points (X₁, Y₁) and (X₂, Y₂) is given by:

S = √ [(X₂ – X₁)² + (Y₂ – Y₁)² ]

For length of line OA, we substitute the above coordinates values and get:

OA = √ [(- 1) – 2)² + (- 1 – 3)² ]

= √ [(- 3)² + (- 4)² ]

= √ ( 9 + 16 )

= √ (25) = 5 units

Therefore, the radius of the circle is 5 units.

Check: Length of diameter = length of A(-1, -1) and B(5, 7)

AB =

AB = = 10 units

Since diameter is 10 units, hence radius will be 5 units

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